# Signals and Systems (GPS and Galileo Receiver) Part 1

The concepts of a signal and a system are crucial to the topic. We will consider time as well as frequency domain models of the signals. We focus on signals and system components that are important to study software-defined GPS and Galileo receiver design.

## Characterization of Signals

In satellite positioning systems, we encounter two classes of signals referred to as deterministic and random signals. Deterministic signals are modeled by explicit mathematical expressions. The signalsare examples of deterministic signals. A random signal, on the other hand, is a signal about which there is some degree of uncertainty. An example of a random signal is a received GPS signal: the received signal contains beside the information bearing signal also noise from disturbances in the atmosphere and noise from the internal circuitry of the GPS receiver.

Now some basic topics on deterministic and stochastic signal theory are reviewed and simultaneously we establish a notation.

A reader familiar with random processes knows concepts like autocorrelation function, power spectral density function (or power spectrum), and bandwidth. These concepts can be applied for deterministic signals as well, and that is exactly what we intend to do in the following.

### Continuous-Time Deterministic Signals

Let us consider a deterministic continuous-time signal x(t), real- or complex-valued with finite energy defined asThe symbol denotes the absolute value, or magnitude, of the complex quantity. In the frequency domain this signal is represented by its Fourier transform:

whereand the variabledenotes angular frequency. By definitionand the units forand f are radian and cycle, respectively. In general, the Fourier transform is complex:

The quantityis often referred to as the spectrum of the signal x (t) because the Fourier transform measures the frequency content, or spectrum, of x(t). Similarly, we refer toas the magnitude spectrum of x (t), and to arg= arctanas the phase spectrum of x (t). Moreover, we refer to as the energy density spectrum of x (t) because it represents the distribution of signal energy as a function of frequency. It is denoted The inverse Fourier transform x(t) ofis

We say that x (t) andconstitute a Fourier transform pair:

The energy density spectrumof a deterministic continuous-time signal x (t) can also be found by means of the (time-average) autocorrelation function (ACF) of the finite energy signal x (t). Let * denote complex conjugation, and then the ACF of x (t) is defined as

is defined as and the energy density spectrum is defined as

Again, we say thatconstitute a Fourier transform pair:

## Discrete-Time Deterministic Signals

Let us suppose that x(n) is a real- or complex-valued deterministic sequence, where n takes integer values, and which is obtained by uniformly sampling the continuous-time signal x(t); read Section 1.2. If x(n) has finite energy

then it has the frequency domain representation (discrete-time Fourier transform)

or equivalently

It should be noted that X (f) is periodic with a period of one andis periodic with a period of

The inverse discrete-time Fourier transform that yields the deterministic sequence x(n) fromor X (f) is given by

Notice that the integration limits are related to the periodicity of the spectra. We refer toas the energy density spectrum of x(n) and denote it as

The energy density spectrumof a deterministic discrete-time signal x(n) can also be found by means of the autocorrelation sequence

via the discrete-time Fourier transform

That is, for a discrete-time signal, the Fourier transform pair is

## Unit Impulse

In signal analysis a frequently used deterministic signal is the unit impulse. In continuous time the unit impulse, also called the delta function, may be defined by the following relation:

where x (t) is an arbitrary signal continuous at t = 0. Its area is

FIGURE 1.1. Rectangular pulse.

In discrete time the unit sample, also called a unit impulse sequence, is defined as

It follows that a continuous-time signal x (t) may be represented as

Similarly, a sequence x(n) may be represented as

The Fourier transform of the unit impulseis given by

which gives us the following Fourier transform pair:

The spectrum of the unit sample is obtained by

which gives us the following Fourier transform pair:

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